Answer
$$\lim\limits _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)=1
$$
The function is continuous at $ y=1$.
Work Step by Step
Given $$\lim\limits _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)
$$
So,
\begin{aligned}a) L&=\lim\limits _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)\\
&= \sec \left( \sec ^{2} 1-\tan ^{2} 1-1\right)\\
&\ \text{since} \ \sec ^{2} y-\tan ^{2}y=1, \text{we get}\\
L&= \sec \left(1-1\right), \\
&= \sec 0\\
&=1\\
\end{aligned}
Since $$ f(y)=\sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)$$
\begin{aligned}b) f(1)&= \sec \left(\sec ^{2} 1-\tan ^{2} 1-1\right)\\
&= \sec0\\
&= 1
\end{aligned}
From (a), (b) since $\lim \limits_{t \rightarrow 1} f(y)=f(1)=1,$ the function is continuous at $ y=1$.
